Generators for automorphisms of special groups

TL;DR

This paper proves that the outer automorphism groups of special groups are finitely generated, describes their generators, and introduces a hierarchical shortening argument to analyze their structure.

Contribution

It establishes finite generation of Out(G) for special groups, identifies the role of poison subgroups, and extends results to coarse-median preserving automorphisms.

Findings

Out(G) is finitely generated for special groups

Dehn twists and pseudo-twists form a virtual generating set

Poison subgroups can be removed by finite-index subgroup replacement

Abstract

Let $G$ be a (compact) special group in the sense of Haglund and Wise. We show that ${\rm Out}(G)$ is finitely generated, and provide a virtual generating set consisting of Dehn twists and ``pseudo-twists''. We exhibit instances where Dehn twists alone do not suffice and completely characterise this phenomenon: it is caused by certain abelian subgroups of $G$, called ``poison subgroups'', which can be removed by replacing $G$ with a finite-index subgroup. Similar results hold for coarse-median preserving automorphisms, without the pathologies: For every special group $G$, the coarse-median preserving subgroups ${\rm Out}(G,[\mu])\leq{\rm Out}(G)$ are virtually generated by finitely many Dehn twists with respect to splittings of $G$ over centralisers. Proofs are based on a novel, hierarchical version of Rips and Sela's shortening argument.